Projective cone

A projective cone (or just cone) in projective geometry is the union of all lines that intersect a projective subspace "R" (the apex of the cone) and an arbitrary subset "A" (the basis) of some other subspace "S", disjoint from "R".

In the special case that "R" is a single point, "S" is a plane, and "A" is a conic section on "S", the projective cone is a conical surface; hence the name.

Definition

Let "X" be a projective space over some field "K", and "R", "S" be disjoint subspaces of "X". Let "A" be an arbitrary subset of "S". Then we define "RA", the cone with top "R" and basis "A", as follows :
* When "A" is empty, "RA" = "A".
* When "A" is not empty, "RA" consists of all those points on a line connecting a point on "R" and a point on "A".

Properties

* As "R" and "S" are disjoint, one easily sees that every point on "RA" not in "R" or "A" is on exactly one line connecting a point in "R" and a point in "A".
* ("RA")$cap$ "S" = "A"
* When "K" = GF("q"), $|R\; A|$ = $q^\{r+1\}$$|A|$ + $frac\{q^\{r+1\}-1\}\{q-1\}$.

ee also

*Cone (geometry)
*Cone (topology)
*Conic section
*Ruled surface
*Hyperboloid